Tagged Questions
2
votes
1answer
51 views
Limit of Binomial distribution
In showing us that Binomial distribution: $$B_{N,p}(n) := \binom {N}{n} p^n(1-p)^{N-n}$$ tends to Poisson's: $$P_ \lambda (n) = \dfrac {\lambda ^n}{n!}e^{-\lambda}$$where I guess lambda should be ...
1
vote
2answers
75 views
Convergent sequences and proof
Prove that $\dfrac{1+n}{n^2}$ converges as $n \to \infty$
How do I go about constructing this proof? Can I use the definition that $\operatorname{abs}(a_n - L < \epsilon)$?
5
votes
4answers
124 views
Proving that $\lim_{h\to 0 } \frac{b^{h}-1}{h} = \ln{b}$
Is there a formal proof of this fact without using L'Hôpital's rule? I was thinking about using a proof
of this fact:
$$
\left.\frac{d(e^{x})}{dx}\right|_{x=x_0} = e^{x_0}\lim_{h\to 0} ...
5
votes
2answers
142 views
Prove $\lim \limits_{x \to\infty } \frac{f(x)}{x}=0$ and $f$ differentiable implies $ \lim \limits_{x \to\infty } \inf |f'(x)|=0 $
Given a differentiable function on $(a,+\infty)$ such as $\lim \limits_{x \to\infty } \frac{f(x)}{x}=0$ prove the following:
$$ \lim \limits_{x \to\infty } \inf |f'(x)|=0 $$
I just can't see how to ...
1
vote
4answers
83 views
How can I prove that a sequence has a given limit?
For example, let's say that I have some sequence $$\left\{c_n\right\} = \left\{\frac{n^2 + 10}{2n^2}\right\}$$ How can I prove that $\{c_n\}$ approaches $\frac{1}{2}$ as $n\rightarrow\infty$?
I'm ...
0
votes
2answers
133 views
$\epsilon$-$\delta$ proof that $f(x) = x^3 /(x^2+y^2)$, $(x,y) \ne (0,0)$, is continuous at $(0,0)$
I need to prove that $f$ continuous at $(x, y)=(0,0)$ using a $\epsilon$-$\delta$ proof
$$
f(x, y) = \begin{cases}
\frac{x^3}{{x^2 + y^2}},&(x,y)\neq (0,0)
\\
0,&(x,y) = (0,0)
\end{cases}
...
1
vote
3answers
66 views
Proving a limit $\lim\limits_{x\to \infty}\frac{x-3}{x^2 +1}$ as $x$ goes to infinity using $\epsilon-\delta$
Hi i need to prove that $$\lim_{x\to \infty}\frac{x-3}{x^2 +1}=0$$ using the formal definition of a limit. Can anyone help?
0
votes
0answers
44 views
Question regarding a Limit involving Logs.
Suppose $A$ is a positive integer and $\delta>0$. Assume the $(n_i)$ are the denominators of a continued fraction representation of the irrational number $[0,a_1,a_2\ldots],$ where $a_{k+1}=A$ ...
9
votes
3answers
198 views
Problem of limit with binomial coefficients
I thought that the following would made a nice exercise, but I am not sure how to evaluate its difficulty since I often miss elementary solutions. How about you try answering it? It would be great to ...
3
votes
1answer
117 views
A Question on Using a Half-Divergent Sequence.
$\theta$ is an irrational in $[0,1]$ with continued fraction representation $[0;a_1,a_2,\dots]$, and the sequences $(a_k), (n_k)$ are related by the recurrence relation $n_{k+1}=a_{k+1}n_k+n_{k-1}, ...
1
vote
1answer
87 views
A changing epsilon for a sequence
Okay so we all know the epsilon-N argument for convergence of sequences, that is a sequence $a_n$ converges to $a$ if $\forall \epsilon > 0, \exists N \in \mathbb{N} : n > N \implies |a_n - a| ...
0
votes
1answer
38 views
Analysis of a limit
I believe I understand this question but I am stuck at what seems to be a "last part."
Here is the question: Suppose that the function $f: \mathbb{R} \rightarrow \mathbb{R}$ is differentiable at ...
0
votes
2answers
113 views
What are common methods/techniques can be used to prove that limit of an infinite sequence exists?
I would like to know what are common methods can be used to show that an infinite sequence converges. From what I know so far,
If a sequence is bounded and monotonic increasing/decreasing then it ...
1
vote
4answers
352 views
Proving $\lim_{x \to \infty} \sqrt{1+4x+x^{2}}-x=2$ [duplicate]
Possible Duplicate:
Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt\[n\]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$
Consider the limit $$\lim_{x \to \infty} \sqrt{1+4x+x^{2}}-x$$
...
1
vote
1answer
48 views
Did I prove this limit correctly?
Given $\lim_{n \to \infty}a_n = L$ and $\lim_{n \to \infty}b_n = M$ implies that $\lim_{n \to \infty}2a_n + 3b_n = 2L + 3M$
Proof
Assume $\lim_{n \to \infty}a_n = L$ and $\lim_{n \to ...
2
votes
2answers
129 views
Is this a valid alternative proof of the sum law in limits?
So sum law of limits tell us
$\lim_{n\to\infty} (a_n+b_n)=X + Y$ if $\lim_{n\to\infty} a_n = X$ and $\lim_{n\to\infty} b_n = Y$
Here is my attempt to prove it.
Proof
Let ...
3
votes
2answers
181 views
Did I underestimate the limit proof?
This is the problem:
Prove that if $a_n \le b_n$ for $n \ge 1, L = \lim_{n \to \infty} a_n$
and $M = \lim_{n \to \infty} b_n$, then $L \le M$
EDIT: Progress
Proof
Assume $L >M$ ...
2
votes
3answers
107 views
Three Limits of Sequences
I'm interested in the proofs of the following: Using the definition of limit (for sequences).
I proved the first one, as shown, but I don't know how to go about doing the second and third.
...
0
votes
0answers
37 views
proving existence of an equilibrium in an equation involving PDF and CDF
I have an equation for which I want to show existence of at least one equilibrium. The equation, call it $h(x)$, is:
$$h(x)=[a-q'(y)]f(x-y) - q''(y)[1-F(x-y)],$$
where $a$ is a positive constant, ...
1
vote
2answers
128 views
How to use the Mean Value Theorem to prove the following statement:
Suppose $f(x)$ is continuous on $[a,b)$ and differentiable on $(a,b)$ and that $f '(x)$ tends to a finite limit $L$ as $x \to a^+$. Then $f(x)$ is right-differentiable at $x=a$ and $f '(a)=L$.
...
3
votes
5answers
164 views
Something is wrong with this proof, limits
Could someone please tell me what is wrong with this proof?
Show that $\lim\limits_{(x,y) \to (0,0)} \dfrac{xy^3}{x^4 + 3y^4}$ does not have a limit or show that it does and find the limit.
I ...
4
votes
2answers
962 views
Continuity proof.
I want to prove that $\exp x$ and $\sin x$ are continuous. This means I want to show that
$$\lim\limits_{x\to a}e^x=e^a$$
$$\lim\limits_{x\to a}\sin x=\sin a$$
for any fixed $a \in \Bbb R$. Then I ...
1
vote
1answer
256 views
Showing that $\cos n\pi\theta$ is periodic unless $\theta$ is an even integer.
I am trying to show that $\cos n\pi\theta$ is periodic unless $\theta$ is an even integer.
I wish to provide a proof based on an example of the $\sin n\pi\theta$ case of the first result, I would ...
3
votes
2answers
1k views
Problem evaluating limits with the variable in the exponent
I have problem evaluating limits with the variable in power, like the following limits:
$\lim_{x \to 0} (1+ \sin 2x)^{\frac{1}{x}}$
$\lim_{x \to \infty} \big(\frac{2x+5}{2x-1})^{2x}$
I asked the ...
5
votes
2answers
779 views
$\epsilon$-$\delta$ limits of functions question
I'm studying how to write epsilon-delta proofs for limits of sequences, limits of functions, continuity, and differentiability and I'm having trouble with the general methodological procedure used in ...
11
votes
8answers
994 views
How to prove that $\lim\limits_{n \to \infty} \frac{k^n}{n!} = 0$
It recently came to my mind, how to prove that the factorial grows faster than the exponential, or that the linear grows faster than the logarithmic, etc...
I thought about writing:
$$
a(n) = ...
14
votes
7answers
918 views
Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.
Why is
$$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$
Can we generalize it to any exponent $x \in \Bbb R$? This is to say, is
$$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$
This is ...
